GWs energy and aLIGO detection

[RockyMarciano]

Yes, I understand that, but how do you know its value?

The answer I get from looking at the reference you gave is this: we know the actual spindown rate of the pulsar from measurements of its pulse period and how that changes over time. So we know the actual rate of rotational energy loss. The question is, how much of that energy loss is due to GWs, and how much is due to EM radiation (which is basically the only other energy loss mechanism available)?

To know that, we have to know the actual EM dipole moment and gravitational quadrupole moment of the pulsar. However, we have no independent way of measuring either of those things. The best we can do is to use the intensity of the EM radiation we detect from the pulsar (namely, its pulses) to estimate how much energy it is radiating as EM waves, and to try to detect GWs from it to get an estimate how much energy it is radiating as GWs.

As I understand it, the EM intensity we see is consistent with all of the observed rotational energy loss being due to EM wave emission. This would indicate that the pulsar is not radiating GWs to any significant extent, and therefore has a negligible gravitational quadrupole moment (or more precisely a negligible third time derivative of same). This is also consistent with the fact that we have observed no GWs from the pulsars in question.

How does all this relate to the quadrupole formula? Well, the quadrupole formula predicts that a negligible third time derivative of the quadrupole moment gives a negligible energy loss due to GW emission. That's why we deduce a negligible third time derivative of the quadrupole moment based on the fact that all of our observations indicate a negligible energy loss due to GW emission. I assume that $\epsilon$ is basically equivalent to "third time derivative of the quadrupole moment" (the latter is what I'm used to seeing in GW emission power formulas), so these observations would also predict a negligible $\epsilon$.
This doesn't seem to be stated correctly (as mfb has pointed out several times now). The difference between the spindown limit prediction for pulsar GW emission, and the actual observed pulsar GW emission, is in the value of $\epsilon$ we plug into the formula; the formula itself is the same in both cases. For the spindown limit, we use the observed rate of energy loss to infer what the value of $\epsilon$ would have to be to account for all of that energy loss being due to GWs. To match the actual observations, we calculate what the upper limit on $\epsilon$ would be such that the energy loss rate due to the GW quadrupole formula is less than the smallest value we could have measured, i.e., the upper limit on $\epsilon$ consistent with no GWs actually being observed. But whatever that rate is, it is exactly the rate given by the quadrupole formula for that value of $\epsilon$. There isn't a different formula for the spindown limit case vs. the actual case.

Thanks for the effort to understand. You are basicallly right and I can see now where mfb and I were talking at cross-purposes. But we all actually agree on the math.

I thought he didn't distinguish between the spindown limit and the actual emission, and I admit I might have misled when talking about what I mean by the quadrupole formula. My claim was that upon the assumption that all the rotational energy loss was converted to GWs the quadrupole formula gave the spindown limit, not that the quadrupole formula cannot be applied by tuning the ellipticity according to observation to compute the actual emission.
That's what the tunable parameter $\epsilon$ does. $\epsilon$ is defined from components of the quadrupole moment $\frac{I_{xx}-I_{yy}}{I_{zz}}$ and it is $7.5*10^{-4}$ for the spindown limit case for Crab pulsar, but as I said one can lower it according to the model of pulsar or the observations from the LIGO interferometer. So yes the formula is in that sense the same, the difference comes from the ellipticity one inserts(for a given $I_{zz}$ that in the the Crab case is $10^{38} Kg.m^2$) and the particular spindown observed.

Now that we are on the same page maybe we can leave behing the prejudices and move on so I can formulate my question?

 

[PeterDonis]

so I can formulate my question?

Your question in the OP, as I understand it, is basically, if GW emission isn't what is causing the spindown of single pulsars, what is causing it, correct?

I have not looked in detail at the numbers, but the obvious candidate to me would be EM radiation emission.